# American Institute of Mathematical Sciences

April  2015, 11(2): 631-644. doi: 10.3934/jimo.2015.11.631

## A penalty-based method from reconstructing smooth local volatility surface from American options

 1 China Center for Special Economic Zone Research, Shenzhen University, 3688 Nanhai Ave., Shenzhen, 518060, China 2 Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, WA 6845, Australia

Received  November 2013 Revised  May 2014 Published  September 2014

This paper is devoted to develop a robust penalty-based method of reconstructing smooth local volatility surface from the observed American option prices. This reconstruction problem is posed as an inverse problem: given a finite set of observed American option prices, find a local volatility function such that the theoretical option prices matches the observed ones optimally with respect to a prescribed performance criterion. The theoretical American option prices are governed by a set of partial differential complementarity problems (PDCP). We propose a penalty-based numerical method for the solution of the PDCP. Typically, the reconstruction problem is ill-posed and a bicubic spline regularization technique is thus proposed to overcome this difficulty. We apply a gradient-based optimization algorithm to solve this nonlinear optimization problem, where the Jacobian of the cost function is computed via finite difference approximation. Two numerical experiments: a synthetic American put option example and a real market American put option example, are performed to show the robustness and effectiveness of the proposed method to reconstructing the unknown volatility surface.
Citation: Kai Zhang, Kok Lay Teo. A penalty-based method from reconstructing smooth local volatility surface from American options. Journal of Industrial & Management Optimization, 2015, 11 (2) : 631-644. doi: 10.3934/jimo.2015.11.631
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